Bayesian belief networks (BBNs) are probabilistic models used to reason under uncertainty. Bayesian belief networks have been successfully used to reason about a wide variety of phenomena, including but not limited to computer vision, social networks, human cognition, and disease detection.
The use of BBNs to provide some computational reasoning capability requires the specification of a priori probability distributions for each node in the network. In applications where there is sufficient prior data, generating these probability distributions can be straightforward. In other applications, various learning algorithms can be used to start from initial distributions and adapt them as a function of incoming data. In many applications, however, BBNs are used to represent expert knowledge or reasoning. In these cases, the construction of a priori probability distributions is substantially more challenging.
A Bayesian belief network includes nodes that are connected by directed edges or links. Each node represents a particular random variable having a certain number of states or values. Each link is directed from a parent node to a child node and shows the causal influence of the parent node on the child node. In particular, the link from a parent node to a child node represents a causal relationship between an event that occurred earlier, as indicated by the state of the parent node, and an event that occurred later, as indicated by the state of the child node.
Every child node in a belief network has an associated conditional probability distribution that describes the causal influence of its parents. The conditional probability distribution of a child node specifies one probability distribution for each combination of values of the parents of the child node.
When all the nodes of a belief network are discrete, each node has a conditional probability table (CPT) associated with it that quantifies the causal probabilistic relationship between that node and its parent nodes, i.e. the a prior probabilities. Using CPTs, and possibly evidence, beliefs can be computed for the nodes of the belief network. Beliefs represent conclusions that can be drawn about the present, using information about the past stored in the CPTs, and using information about the present stored in evidence, if any. A belief for a node X represents a conditional probability distribution of the node X, given all available evidence for that node.
To compute beliefs using a belief network that has discrete nodes, users of the belief network typically enter the CPT values for each node, based on the number of states of that node and on the number of parents that the node has. Such a process can become unwieldy, because the number of CPT values that must be specified for a node increases exponentially with the number of states and parents of the node. The CPT specifies one probability distribution over the states of the child node for each combination of states of its parent nodes. The number of these distributions in the child node's CPT grows exponentially in the number of parents and the number of states per parent and can quickly exceed a reasonable number. The potential for this exponential explosion can reduce the applicability of BBNs to many problem domains. This exponential explosion, combined with the underlying sophistication of the representation, presents a challenge.
A canonical model makes a specific assumption about the type of relationship between a node and its parents. This assumption results in many fewer parameters being needed to specify an entire CPT. There are many types of canonical models used in practice and each assumes a different relationship.
In the parent '085 application, one type of canonical model called the Causal Influence Model (CIM) was disclosed. The CIM uses a linear function to combine parent influences, resulting in belief values that are perceived as more intuitive by users, regardless of the input parameters. The CIM assumes that each node is discrete and has an arbitrary number of states with arbitrary meaning. Each node has a baseline probability distribution, independent of any parent effects. Each parent independently influences these baseline probabilities to be more or less likely.
The Causal Influence Model (CIM) provides an intuitive way to reduce the number of parameters required for a Conditional Probability Table (CPT) from an exponential function of the number of parents to a linear function of the number of parents. However, for each parent Xi the user must still specify a matrix of parameters, called the causal influence matrix. The matrix has a size mi×m, where mi is the number of states of parent Xi and m is the number of states of child Y. For each parent, the user must specify a number of influences equal to the product of the number of parent states and child states for the CIM. While this is certainly better than the CPT, it can still result in too many parameters for a user to specify.
Methods and systems that simplify and reduce the number of parameters needed to specify the causal influence matrix are therefore desirable.